Found problems: 471
An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.
Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$.
[i]Proposed by Evan Chen[/i]
For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$
Consider the sequence in which $a_1 = 1$ and $a_n$ is obtained by juxtaposing the decimal representation of $n$ at the end of the decimal representation of $a_{n-1}$. That is, $a_1 = 1$, $a_2 = 12$, $a_3 = 123$, $\dots$ , $a_9 = 123456789$, $a_{10} = 12345678910$ and so on. Prove that infinitely many numbers of this sequence are multiples of $7$.
Which of the following is equivalent to $$(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$$
$\textbf{(A) }3^{127}+2^{127} \qquad \textbf{(B) }3^{127}+2^{127}+2\cdot 3^{63}+3\cdot 2^{63} \qquad \textbf{(C) }3^{128}-2^{128} \qquad \textbf{(D) }3^{128}+2^{128} \qquad \textbf{(E) }5^{127}$
Let $\mathbb{N}=\{1, 2, 3, \dots\}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold:
(1) $f(ab) = f(a)f(b)$, and
(2) at least two of the numbers $f(a)$, $f(b)$, and $f(a+b)$ are equal.
Determine the greatest common divisor of the elements of the set \[\{n^{13}-n \; \vert \; n \in \mathbb{Z}\}.\]
Show that there exist infinitely many composite positive integers $n$ such that $n$ divides $2^{\frac{n-1}{2}}+1$
Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$.
[i]Proposed by Evan Chen[/i]
Suppose $a_1,a_2,a_3,b_1,b_2,b_3$ are distinct positive integers such that
\[(n \plus{} 1)a_1^n \plus{} na_2^n \plus{} (n \minus{} 1)a_3^n|(n \plus{} 1)b_1^n \plus{} nb_2^n \plus{} (n \minus{} 1)b_3^n\]
holds for all positive integers $n$. Prove that there exists $k\in N$ such that $ b_i \equal{} ka_i$ for $ i \equal{} 1,2,3$.
Let $p > 3$ be a prime. Prove that if \[ \sum_{i=1 }^{p-1}{1\over i^p} = {n\over m}, \] with $\gdc(n,m) = 1$, then $p^3$ divides $n$.
For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1, a_2, . . .$ and an infinite geometric sequence of integers $g_1, g_2, . . .$ satisfying the following properties?
[list]
[*] $a_n - g_n$ is divisible by $m$ for all integers $n \ge 1$;
[*] $a_2 - a_1$ is not divisible by $m$.
[/list]
[i]Holden Mui[/i]
We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$.
Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$.
Is there always a number $x$ that satisfies all the equations?
[i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]
Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$.
[i]Victor Wang[/i]
Let $a(n)$ be the sequence defined by $a(1)=2$ and $a(n+1)=(a(n))^{n+1}-1$ for each integer $n\geq 1$. Suppose that $p>2$ is a prime and $k$ is a positive integer. Prove that some term of the sequence $a(n)$ is divisible by $p^k$.
[i]Proposed by John Berman[/i]
Let $\mathbb{P}$ be the set of all primes and let $M$ be a subset of $\mathbb{P}$ with at least three elements.
Suppose that for all $k \geq 1$ and for all subsets $A=\{p_1,p_2,\dots ,p_k \}$ of $M$ ,$A\neq M$ , all prime factors of $p_1p_2\dots p_k-1$ are in $M$ . Prove that $M=\mathbb{P}$.
Let $N$ be the number of polynomials $P(x_1, x_2, \dots, x_{2016})$ of degree at most $2015$ with coefficients in the set $\{0, 1, 2 \}$ such that $P(a_1,a_2,\cdots ,a_{2016}) \equiv 1 \pmod{3}$ for all $(a_1,a_2,\cdots ,a_{2016}) \in \{0, 1\}^{2016}.$
Compute the remainder when $v_3(N)$ is divided by $2011$, where $v_3(N)$ denotes the largest integer $k$ such that $3^k | N.$
[i]Proposed by Yang Liu[/i]
For any positive integer $a,$ $\sigma(a)$ denotes the sum of the positive integer divisors of $a.$ Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a.$ Find the sum of the prime factors in the prime factorization of $n.$
Show that there do not exist polynomials $p(x)$ and $q(x)$ each having integer coefficients and of degree greater than or equal to 1 such that \[ p(x)q(x) = x^5 +2x +1 . \]
Find all pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$
Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not.
[i](Russia) Valery Senderov[/i]
Find the number of primes $p$ between $100$ and $200$ for which $x^{11}+y^{16}\equiv 2013\pmod p$ has a solution in integers $x$ and $y$.
Find all pairs $(p,q)$ of prime numbers which $p>q$ and
$$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$
is an integer.
Find all prime numbers $p,q$ for which $pq$ divides $(5^p-2^p)(5^q-2^q)$.
Let $k\geq 2$,$n_1,n_2,\cdots ,n_k\in \mathbb{N}_+$,satisfied $n_2|2^{n_1}-1,n_3|2^{n_2}-1,\cdots ,n_k|2^{n_{k-1}}-1,n_1|2^{n_k}-1$.
Prove:$n_ 1=n_ 2=\cdots=n_k=1$.